powers and exponent laws - ms. johnsonsolve problèms involving powers. • perform operations with...

Powers and Exponent Laws

Imagine folding a biece of paper in half to form 2 I; lyers.

Imagine folding it in half again to form 4 layers.

Try this with a sh How many times paper before it is make another folc

elet of paper, ^an you fold the mpossible to

What You'll Learn

Us e powers to represent repeated multiplication.

Use pat ;erns to understand a power with expqnent 0.

Solve problèms involving powers. • Perform operations with powers.

• Explain and apply the order of operations with exponents.

Why It's Important Powers provide an efficient ' to record our work. The prof of powers lead to even more efftè ways to perform some calculation Powers are used in many formula: applications in science, constructic design.

50

2.1 What Is a Power?

«" viV.f ^ l ' "Jf. *V r

FOCUS • Use powers to

represent repeated multiplication.

What is the area of this square? Write the area as a product.

What is the volume of this cube? Write the volume as a product.

3 units

1 unit

I n v e s t i g a t e « © t

You will need congruent square tiles and congruent cubes. >- Use the tiles to make as many different-sized larger squares as you can.

Write the area of each square as a product. Record your results in a table.

Number of

1

Tiles Area (square units) Side Length (units) Area as a fjroduct

1x1

>- Use the cubes to make as many different-sized larger cubes as you can. Write the volume of each cube as a product. Record your results in a table.

Number of tubes of ju Volume (cubic units) Edge Length (units) Volume as a product

1x1x1

What patterns do you see in the tables? Use the patterns to predict the areas of the next 3 squares and the volumes of the next 3 cubes. How are these areas and volumes the same? How are they different?

5 2 UNIT 2: Powers and Exponent Laws

C o n n e c t

When an integer, other than 0, can be written as a product of equal factors, we can write the integer as a power. For example, 5 X 5 X 5 is 53. 5 is the base. 3 is the exponent. ( c 3 53 is the power.

53 is a power of 5.

base

exponent

power

We say: 5 to the 3rd, or 5 cubed

>- A power with an integer base and exponent 2 is a square number. When the base is a positive integer, we can illustrate a square number.

Here are 3 ways to write 25. Standard form: 25 As repeated multiplication: 5 X 5 As a power: 52

5 x 5 = 52

= 25 25 is a square number.

>• A power with an integer base and exponent 3 is a cube number. When the base is a positive integer, we can illustrate a cube number.

5 5 Here are 3 ways to write 125.

Standard form: 125 As repeated multiplication: 5 X 5 X 5 5 As a power: 53

Writing Powers

5 x 5 x 5 = 53

= 125 125 is a cube number.

Write as a power, a) 3 X 3 X 3 X 3 X 3 X 3

•/I Solution

b)7

a) 3 X 3 X 3 X 3 X 3 X 3 The base is 3. There are 6 equal factors, so the exponent is 6. So, 3 X 3 X 3 X 3 X 3 X 3 = 36

b) 7 The base is 7. There is only 1 factor, so the exponent is 1. So, 7 = 71

2.1 W h a t Is a Power? 5 3

Evaluating Powers

Write as repeated multiplication and in standard form, a) 35 b) 74

M Solution

a) 35 = 3 X 3 X 3 X 3 X 3 As repeated multiplication = 243 Standard form

b) 74 = 7 X 7 X 7 X 7 As repeated multiplication = 2401 Standard form

Examples 1 and 2 showed powers with positive integer bases. A power can also be negative or have a base that is a negative integer.

M M M i M Evaluating Expressions Involving Negative Signs

Identify the base of each power, then evaluate the power, a) (—3)4 b) - 3 4 c) —(—34)

• A Solution

a) The base of the power is —3. ( —3)4 = ( - 3 ) X ( - 3 ) X ( - 3 ) X ( - 3 ) Apply the rules for multiplying integers: The sign of a product with an even number of negative factors is positive. So, (—3)4 = 81

b) The base of the power is 3. The exponent applies only to the base 3, and not to the negative sign. - 3 4 = — (34)

= - ( 3 X 3 X 3 X 3) = - 8 1

c) From part b, we know that — 34 = —81. So, - ( - 3 4 ) = - ( - 8 1 )

= 81

As repeated multiplication

Standard form

— (—81) is the opposite of —81, which is 81.


We may write the product of integer factors without the multiplication sign. In Example 3a, we may write (—3) X ( — 3) X ( — 3) X (—3) as ( — 3)(—3)(—3)( — 3).

A calculator can be used to evaluate a power such as (—7)5

in standard form. r - n ~ r _

- IJ D = i c p m

- l O O U I.

1. Can every integer, other than 0, be written as a power? Explain.

2. Why is — 34 negative but (—3)4 positive? Give another example like this.

3. Two students compared the calculator key sequences they used to evaluate a power. Why might the sequences be different?

Practice

Check 4. Write the number of unit squares in each

large square as a power. a) b)

5. Write the number of unit cubes in each large cube as a power, a) b) c)

6. Use grid paper. Draw a picture to represent each square number. a) 42 b) 6 X 6 c) 49

d) 102 e) 81 f) 12 X 12

7. Write the base of each power, a) 2 7 b) 43

c) 8 2 d) ( - 1 0 ) 5

e) ( - 6 ) 7 f) - 8 3

8. Write the exponent of each power, a) 25 b) 6 4

c) 9 1 d) — 32

e) ( - 2 ) 9 f) (—8)3

9. Write each power as repeated multiplication. a) 32 b) 104

c) 8 s d) (—6)5

e) — 6 5 f) - 4 2

10. a) Explain how to build models to show the difference between 32 and 23.

b) Why is one number called a square number and the other number called a cube number?


11. Use repeated multiplication to show why 64

is not the same as 46.

12. Write as a power. a) 4 X 4 X 4 X 4

b) 2 X 2 X 2

c) 5 X 5 X 5 X 5 X 5 X 5

d) 10 X 10 X 10

e) ( - 7 9 X - 7 9 )

f) - ( - 2 ) ( - 2 ) ( - 2 ) ( - 2 ) ( - 2 ) ( - 2 ) ( - 2 ) ( - 2 )

Apply 13. Write each product as a power, then evaluate.

a) 5 X 5

b) 3 X 3 X 3 X 3

c) 10 X 10 X 10 X 10 X 10

d) - ( 9 X 9 X 9)

e) ( - 2 X - 2 X - 2 )

0 - ( - 4 ) ( - 4 ) ( - 4 )

g) ( - 5 X - 5 X - 5 X - 5 ) h) - ( 5 ) ( 5 ) ( 5 ) ( 5 )

i) - ( - 5 X - 5 X - 5 X - 5 )

14. Predict whether each answer is positive or negative, then evaluate. a) 2 3 b) 106 c) 3 1 d) - 7 3

e) ( - 7 ) 3 f) ( - 2 ) 8 g) - 2 8 h) ~64

i) ( - 6 ) 4 j) ~ ( - 6 ) 4 k) ( —5)3 I) — 4 4

15. Canada Post often creates special postage stamps to celebrate important events and honour famous people.

a) Captain George Vancouver was a Dutch explorer who named almost 400 Canadian places. To commemorate his 250th birthday in 2007, Canada Post created a $1.55 stamp.

i) How many stamps are in a 3 by 3 block? Write the number of stamps

j as a power. ii) What is the value of these stamps?

b) In July 2007, Canada hosted the FIFA U-20 World Cup Soccer Championships. Canada Post issued a 52<t stamp to honour all the players and fans.

i) How many stamps are in a 4 by 4 block? Write the number of stamps as a power.

ii) What is the value of these stamps?

16. Evaluate. a) 3 1 2 b) - I 1

c) 511 d) —(—4)10

e) (—9)8 f) 2 2 3

17. Assessment Focus a) Write as repeated multiplication and in

standard form. I i) 43 ii) —43 iii) — (—43) iv) (—43) | b) Which products in part a are positive? ! Why? Which products are negative? Why?

c) Write as repeated multiplication and in standard form. i) 42 ii) — 42 iii) - ( - 4 2 ) iv) ( - 4 2 )

d) Which products in part c are positive? Why? Which products are negative? Why?

e) Write other sets of powers like those in parts a and c. Explain how you know if each product is positive or negative before you write the power in standard form.


18. a) Is the value of —35 different from the value of (—3)5 or (—35)? What purpose do the brackets serve?

b) Is the value of — 46 different from the value of (—4)6 or (—46)? What purpose do the brackets serve?

19. a) When does a negative base in a power produce a negative product? Give 3 examples.

b) When does a negative base in a power produce a positive product? Give 3 examples.

Take It Further 20. Write each number as a power with base 2.

Explain your method. a) 4 b) 16 c) 64

d) 256 e) 32 f) 128

21. a) Write each number as a power in as many ways as possible. i) 16 ii) 81 iii) 256

b) Find other numbers that can be written as a power in more than one way. Show your work.

22. a) How are the powers in each pair the same? How are they different?

i) 23 or 32 ii) 25 or 52

iii) 34 or 4 3 iv) 54 or 45

23. Without evaluating all the powers, write them in order from greatest to least: 35, 52, 34, 63

Explain your strategy.

24.

b) In part a, which is the greater power in each pair? Explain how you know.

How many squares of each side length are there on a checkerboard? Write each number as a power. a) 1 unit c) 3 units e) 5 units g) 7 units

b) 2 units d) 4 units f) 6 units h) 8 units

What patterns do you see in the answers?

25. Explain how to tell if a number is a square number, or a cube number, or neither. Give examples.

What is a power? Why are brackets used when a power has a negative base?


2.2 Powers of Ten and the Zero Exponent

FOCUS • Explore patterns

and powers of 10 to develop a meaning for the exponent 0.

Nuclear reactions in the core of the sun create solar energy. For these reactions to take place, extreme temperature and pressure are needed. The temperature of the sun's core is about 107 °C. What is this temperature in millions of degrees Celsius?

: ' A

Investigate

Choose a number between 1 and 10 as the base of a power. Use the exponents 5, 4, 3, 2, and 1. Use your base and each exponent to write a power. Then write the power as repeated multiplication and in standard form. Record your results in a table.

Exponent

5 4 3 2 1

Describe any patterns in your table. Continue the patterns to complete the entries in the last row.

Repeated ! Standard Multiplication Form

Compare your tables and patterns with those of other pairs of students. What do you think is the value of a power with exponent 0? Use a calculator to check your answer for different integer bases.


Connect

This table shows decreasing powers of 10,

Number iji Words Standard Form

One billion 1 000 000 000 One hundred million 100 000 000 Ten million 10 000 000 One million 1 000 000 One hundred thousand 100 000 Ten thousand 10 000 One thousand 1 000 One hundred 100 Ten 10 One 1

Power

109

108

107

106

105

104

103

102

101

^ We use the pattern in the exponents to write 1 as 10°.

We could make a similar table for the powers of any integer base except 0. So, 1 can be written as any power with exponent 0. For example, 1 = 2 °

1 = 13° 1 = ( - 5 ) °

• Zero Exponent Law A power with an integer base, other than 0, and an exponent 0 is equal to 1. n° = 1, n # 0

Evaluating Powers with Exponent Zero

Evaluate each expression. a) 4° b) - 4 ° c) ( - 4 ) °

• A Solution

A power with exponent 0 is equal to 1.

a ) 4 ° = l b) —4° = — 1 c ) ( - 4 ) ° = l

We can use the zero exponent and powers of 10 to write a number.

2.2 Powers of Ten and t h e Zero Exponent 5 9

mrnamn Write 3452 us ing powers of 10.

M Solution

Use a place-value char t .

Thousands Hundreds Tens Ones

3 4 5 2

3452 = 3000 + 400 + 50 + 2

= (3 X 1000) + (4 X 100) + (5 X 10) + (2 X 1) We use brackets for clarity.

= (3 X 103) + (4 X 102) + (5 X 101) + (2 X 10°)

j M H l j interpreting Numbers in the Media

Head-Smashed-ln Buffalo Jump is a UNESCO World Heritage Site j in Southern Alberta. This site covers 600 hectares and contains

cultural remains used in the communal hunting of buffalo. Head-Smashed-ln was first used for hunting bison at least 5700 years ago and perhaps as early as 10 000 years ago. It is estimated that close to sixty million Plains Bison roamed the prairies prior to the Europeans' • arrival in Western Canada. Less than one hundred years later, fewer than 1000 animals remained. |

1 1' i f l i i i i i i j S M

Use powers of 10 to wr i te each n u m b e r in the above pa rag raph .

M Solution

600 = 6 X 100

= 6 X 102

5700 = 5000 + 700

= (5 X 1000) + (7 X 100)

= (5 X 103) + (7 X 102)

10 000 = 1 X 104

60 000 000 = 6 X 10 000 000

= 6 X 107

100 = 1 X 102

1000 = 1 X 103


1. In Example 1, why are 4° and (—4)° equal to 1, while —4° is equal to — 1?

2. What is meant by "a power of 10"? Name 6 numbers that are powers of 10.

3. How would you use patterns to explain that 10° = 1?

Practice

Check 4. Evaluate each power,

a) 50° b) 9° c) 1° d) 17°

5. Evaluate each power. a) ( - 6 ) ° b) - 1 1 ° c) - 8 ° d) ( - 2 4 ) °

6. Write each number as a power of 10. a) 1000 b) 100 000 c) 1 000 000 000

d) ten thousand e) one hundred billion

Apply 7. Write 1 as a power in three different ways.

8. Evaluate each power of 10. a) 107 b) 102 c) 10°

d) 1010 e) 101 f) 106

9. Use powers of 10 to write each number, a) 6 000 000 000 b) 200 c) 51 415

d) 60 702 008 e) 302 411 f) 2 000 008

10. Write each number in standard form. a) 7 X 107

b) (3 X 104) + (9 X 103) + (5 X 101)

+ (7 X 10°)

c) (8 X 108) + (5 X 105) + (2 X 102)

d) (9 X 1010) + (8 X 109) + (1 X 10°)

e) 1 X 1015

f) (4 X 103) + (1 X 10°) + (9 X 105)

+ (3 X 101)

11. The data below refer to trees in Vancouver. Use powers of 10 to write each number. • Street trees have an estimated value of

over $500 million. • In the past decade, the Park Board has

planted almost 40 000 new street trees. • Nearly 3 million ladybugs are released

every year to help control aphids on street trees.

• The most common street tree is the Japanese flowering cherry, with over 17 000 growing on city streets.

• There are 130 000 trees lining the streets of Vancouver.

• There are nearly 600 different types of trees.

12. Assessment Focus Choose a negative integer as the base of a power. Copy and complete the table below. Use patterns to explain why the power with exponent 0 is equal to 1.

Exponent

5 4 3 2 1

0

Power Standard Form

2.2 Powers of Ten and t h e Zero Exponent 6 1

13. In each pair, which number is greater? How do you know? a) (4 X 103) + (6 X 102) + (6 X 101)

+ (7 X 10°) o r 4327

b) (2 X 104) + (4 X 103) + (2 X 102)

+ (4 X 101) o r 2432

c) (7 X 1()7) + (7 X 103) o r 111 111

• Worldwide, about one billion people lack access to safe drinking water.

• Glacier ice over 100 000 years old can be found at the base of many Canadian Arctic ice caps.

• Approximately 1000 kg of water is required to grow 1 kg of potatoes.

Why is a power with exponent 0 equal to 1?

! • Henderson Lake, British Columbia, has | | the greatest average annual precipitation I in Canada of 6655 mm. That is more |

| than 100 times as much as Eureka, in | Nunavut, which has the least average | annual precipitation of 64 mm. [ • In November 2007, at the request of local ! First Nations, over 10 million hectares of ! the Mackenzie River Basin were protected j from industrial development.

i Using this information: I a) Identify the powers of 10 and write them | using exponents. | b) Arrange the numbers in order from least | to greatest. ! c) Explain how writing powers of 10 using j exponents can help you to order and | compare numbers. i

| Take It Further i 15. What are the meanings of the words trillion, j quadrillion, and quintillion? | Write these numbers as powers. ! What strategies did you use?

Your World The amount of data that an MP3 player can store is measured in gigabytes. For example, one MP3 player can store 2 GB (gigabytes) of songs. One song uses about 7000 KB (kilobytes) of space, where 1 GB = 220 KB. About how many songs can the MP3 player hold?


2.3 Order of Opera t ions with Powers

r FOCUS • Explain and apply the

order of operations with exponents.

This was a skill-testing question in a competition: 6 X (3 + 2) - 10 -h 2

Which answer is correct: 5, 10, 15, or 25? How do you know?

Investigate

Use each of the digits 2, 3, 4, and 5 once to write an expression. >- The expression must have at least one power.

The base of the power can be a positive or negative integer. >• The expression can use any of:

addition, subtraction, multiplication, division, and brackets Evaluate the expression. Write and evaluate as many different expressions as you can.

f p a Share your expressions with another pair of students. Where does evaluating a power fit in the order of operations? Why do you think this is?

2.3 Order of Opera t ions with Powers

Connect

To avoid getting different answers when we evaluate an expression, we use this order of operations:

• Evaluate the expression in brackets first. • Evaluate the powers. • Multiply and divide, in order, from left to right. • Add and subtract, in order, from left to right.

Adding and Subtracting with Powers

Evaluate. a) 3 3 + 23 b) 3 - 23 c) (3 + 2)3

A Solution

a) Evaluate the powers before adding. b) Evaluate the power, then subtract. 33 + 2 3 = (3) (3)(3) + (2)(2)(2) 3 - 23 = 3 - (2) (2)(2)

= 27 + 8 = 3 - 8

= 35 = - 5

c) Add first, since this operation is within the brackets. Then evaluate the power. (3 + 2) 3 = 5 3

= (5) (5) (5)

= 125

When we need curved brackets for integers, we use square brackets to show the order of operations. When the numbers are too large to use mental math, we use a calculator.

/ ' / y m Multiplying and Dividing with Powers Evaluate. a) [2 X ( —3)3 — 6] 2 b) (182 + 5°)2 -5- ( —5)3

• A Solution

a) Follow the order of operations. Do the operations in brackets first: evaluate the power (—3)3

[2 X ( —3)3 - 6]2 = [2 X ( - 2 7 ) - 6]2 Then multiply: 2 X ( - 2 7 ) = [ —54 — 6]2 Then subtract: —54 — 6 = (—60)2 Then evaluate the power: (—60)2

= 3600


b) Use a calculator to evaluate (182 + 5°)2 ( — 5)3. For the first bracket: Use mental math when you can: 5° = 1 Evaluate 182 + 1 to display 325. Evaluate 3252 to display 105 625. For the second bracket: ( —5)3 is negative, so simply evaluate 53 to display 125 To evaluate 105 625 + ( — 125), the integers have opposite signs, so the quotient is negative. Evaluate 105 625 -h 125 to display 845. So, (182 + 5°)2 -h (—5)3 = - 8 4 5

w m m Solving Problems Using Powers

Lyn has a square swimming pool, 2 m deep with side length 4 m. The swimming pool is joined to a circular hot tub, 1 m deep with diameter 2 m. Lyn adds 690 g of chlorine to the pool and hot tub each week. This expression represents how much chlorine is present per 1 m3 of water:

690 2 X 42 + tt X l3

The suggested concentration of chlorine is 20 g/m3 of water. What is the concentration of chlorine in Lyn's pool and hot tub? Is it close to the suggested concentration?

• A Solution

Use a calculator. Since the denominator has a sum, draw brackets around it. This ensures the entire denominator is divided into the numerator. Key in the expression as it now appears: — — 3, — 19.634 85

( 2 X 4 + 7T X 1 j The concentration is about 19.6 g/m3. This is very close to the suggested concentration.

1. Explain why the answers to 33 + 23 and (3 + 2)3 are different.

2. Use the meaning of a power to explain why powers are evaluated before multiplication and division.

182 + l 3c?5.

in 5S25.

5 " 3 155.

105655/155 8H5.

2.3 Order of Opera t ions with Powers 6 5

Practice

Check 3. Evaluate,

a) 32 + 1

c) (3 + l ) 2

e) 22 + 4

g) (2 + 4) 2

i) 2 - 42

b) 32 - 1

d) (3 - l ) 2

f) 2 2 - 4

h) (2 - 4) 2

j) 22 - 42

4. Evaluate. Check using a calculator. a) 2 3 X 5

c) (2 X 5) 3

e) ( - 1 0 ) 3 - 5

g) [ ( - 1 0 ) - 5] 3

5. Evaluate. a) 2 3 + (—2)3

c) 2 3 - ( —3)3

e) 2 3 + ( - 1 ) 3

g) 2 3 X (—2)3

b) 2 X 5 2

d) (2 X 5) 2

f) ( - 1 0 ) - 5°

h) [ ( - 1 0 ) - 5]°

b) (2 - 3) 3

d) (2 + 3) 3

f) (2 - 2) 3

h) (2 X l ) 3

8. State which operation you will do first, then evaluate. a) (7)(4) - (5)2 b) 6(2 - 5)2

c) ( —3)2 + (4)(7) d) ( - 6 ) + 4° X ( - 2 )

e) 102 + [10 - (—2)]2 f) [18 -f- (—6)]3 X 2

9. Sometimes it is helpful to use an acronym . . _ An acronym is

as a memory trick. Create an acronym to a word formed

help you remember the order of operations, from the first

Share it with your classmates. 'etters ot^er

words.

10. Evaluate. a) (3 + 4) 2 X (4 - 6) 3

Apply 6. a) Evaluate. Record your work.

i) 42 + 4 3 ii) 5 3 + 5 6

b) Evaluate. Record your work. i) 6 3 - 62 ii) 6 3 - 6 5

7. Identify, then correct, any errors in the student work below. Explain how you think the errors occurred.

; I 1 +• 7 2 x 2 * (~<°)

j =r <? +- 4 x l 6 - 3 6

I =• 1*5 x \ 6 - 3 6

\n

b) (8

32 -L.

22 + l)3 - 35

c) 4 3 h- [8(6° - 2 1)]

d) [9 - ( - 3 ) ] 2

e) (22 X l3)2

f) ( l l 3 4- 52)0 + (42 - 24)

11. Explain why the brackets are not necessary to evaluate this expression. (—43 X 10) - (6 - 2) Evaluate the expression, showing each step.

12. Winona is tiling her 3-m by 3-m kitchen floor. She bought stone tiles at $70/m2. It costs $60/m2 to install the tiles. Winona has a coupon for a 25% discount off the installation cost. This expression represents the cost, in dollars, to tile the floor: 70 X 32 + 60 X 32 X 0.75 How much does it cost to tile the floor?

13. Evaluate this expression: 23 + (3 X 4)2 - 6 Change the position of the brackets. Evaluate the new expression. How many different answers can you get by changing only the position of the brackets?

6 6 UNIT 2: Powers a n d Exponen t Laws

14. Evaluate each pair of expressions. Why are some answers the same? Why are other answers different? a) 3 + 5 X 8 and 5 X 8 + 3 b) 32 + 22 and (3 + 2)2

c) 33 X 23 - 52 and (3 X 2)3 - 5 X 5 d) 23 X 32 and (2 X 3)5

e) 5 X 3 - 32 X 4 + 20 X 7 and 5 X (3 - 32) X 4 + 20 X 7

15. This student got the correct answer, but she did not earn fall marks. Find the mistake this student made. Explain how it is possible she got the correct answer. Write a more efficient solution for this problem.

! - ( 2 A - ? x 4-2)5 (-2-f j ^

\ =, - ( 1 4 . - 144)° ( -?)

16. Use a calculator to evaluate. Write the key strokes you used. a) ( 1 4 + 1 0 ) 2 X ( 2 1 - 2 8 ) 3

b) ( 3 6 ^ 2 2 + 1 1 ) 3 - 1 0 5

c) 1 2 3

C 3 6 ( 1 2 ° - 1 3 1 )

d) 8 1 2

92 + ( - 9 ) 2

e) ( 1 4 2 + 6 3 ) 2

f) ( 1 1 3 + 2 5 2 ) 0 + ( 2 7 2 - 3 3 4 )

17. Assessment Focus Predict which expression has a value closest to 0. Explain your strategy for predicting, then verify your prediction. (30 + 9 X 11 H- 3 ) °

( - 3 X 6) + 42

1 + ( 1 - 1 ) 2 + 1°

18. Robbie, Marcia, and Nick got different answers when they evaluated this expression: (—6)2 — 2[(—8) + 2]2

Robbie's answer was 68, Marcia's answer was 4, and Nick's answer was —68. a) Who had the correct answer? b) Show and explain how the other two

students might have got their answers. Where did they go wrong?

19. A timber supplier manufactures and delivers wood chips. The chips are packaged in boxes that are cubes with edge length 25 cm. The cost of the chips is $14/m3, and delivery costs $10 per 25 km. One customer orders 150 boxes of wood chips and she lives 130 km from the supplier. This expression represents the cost, in dollars: 1 0 ^ 5

1 3 0 + 2 5 3 h- i o 6 X 1 4 X 1 5 0

How much does the customer pay?

2.3 Order of Opera t ions with Powers 6 7

20. Copy each statement. Insert brackets to make each statement true. a) 10 + 2 X 32 - 2 = 106

b) 10 + 2 X 32 - 2 = 24

c) 10 + 2 X 32 - 2 = 84

d) 10 + 2 X 32 - 2 = 254

21. Copy each statement. Insert brackets to make each statement true. a) 20

b) 20

c) 20

d) 20

2 + 2 X 2 z + 6 = 26

2 + 2 X 2 2 + 6 = 30

2 + 2X22+ 6 = 8

2 + 2 X 22 + 6 = 120

22. Blake answered the following skill-testing question to try to win a prize: 5 X 42 — (23 + 33) - 5

Blake's answer was 11. Did Blake win the prize? Show your work.

23. Write an expression that includes integers, powers, brackets, and all four operations. Evaluate the expression. Ask a classmate to evaluate the expression. Did both of you follow the same order of operations? Is it possible to get the same answer if you follow a different order of operations? Explain.

Take It Further 24. Copy and complete each set of equations.

Describe any patterns you see. Extend each pattern by 2 more rows, a) l 3 = l 2

l 3 + 23 = 32

l 3 + 23 + 3 3 = 6 2

l 3 + 23 + 33 + 4 3 =

l 3 + 2 3 + 3 3 + 4 3 + 5 3 =

b) 32 - l 2 = D 3

62 — 32 = n 3

io2 - 62 = n 3

152 - 102 = D 3

21 2 - 152 = m 3

25. Choose two numbers between —5 and +5. a) Square the numbers, then add the

squares. Write this as an expression. b) Add the numbers, then square the sum.

Write this as an expression. c) Compare the answers to parts a and b.

What do you notice? d) A student said, "The sum of the squares

of two numbers is equal to the square of the sum of the numbers." Do you agree with this statement? Justify your answer.

26. Use four 4s and any operations, brackets, or powers to write an expression for each whole number from 1 to 9.

27. a) Write each product as a power of 2 and in standard form.

i) 2 X 2 X 2 X 2

ii) 2 X 2

iii) 2 X 2 X 2 X 2 X 2

iv) 2 X 2 X 2

b) Write each number as a sum, using only powers of 2. For example: 27 = 1 6 + 8 + 2 + 1

= 24 + 23 + 21 + 2° i) 28 ii) 12

iii) 25 iv) 31

v) 50 vi) 75

c) Repeat part b with a different base. Share your results with a classmate.

Why is the order of operations important? Include examples in your explanation.

6 8 UNIT 2: Powers a n d Exponent Laws

Mid-Unit Review

1. Write each power in standard form, a) 142 b) 5 1 c) —83

d) —(—4)4 e) ( —6)3 f) ( —2)8

2. Copy and complete this table.

Power Base Exponent Repeated Multiplication

a) 43

b) 25

c) 86

d) 7 2

e) 3 x 3 x 3 x 3

Standard Form

3. a) Evaluate the first 8 powers of 7. Copy and complete this table.

Power of 7 Standard Form

71

72

73

74

75

76

V I s

b) What pattern do you see in the ones digits of the numbers in the second column?

c) Verify that the pattern continues by extending the table for as many powers of 7 as your calculator displays.

d) Use the pattern. Predict the ones digit of each power of 7. Explain your strategy.

i) 712 ii) 714

iii) 717 iv) 722

4. Write in standard form. a) 106 b) 10° c) 108 d) 104

5. Write as a power of 10. a) one billion b) one c) 100 d) 100 000

6. Evaluate. a) ( - 5 ) ° b) 25° c) - 6 ° d) 9°

7. The area of land is measured in hectares (ha). One hectare is the area of a square with side length 100 m. Write the number of square metres in 1 ha as a power.

8. Evaluate. State which operation you do first. a) ( - 2 1 - 6) 2 + 14

b) 6 4- ( - 2 ) + (2 X 3)2

c) [5 - (—4)] 3 - (21 - 7) 4

d) [(6 - 21) 3 X (2 + 2 ) 6 ] 0

e) (3 - 5) 5 - ( - 4 )

f) - 3 0 - (7 - 4 ) 3

9. Both Sophia and Victor evaluated this expression: —24 X 5 + 16 (— 2)3

Sophia's answer was —82 and Victor's answer was 78. Who is correct? Find the likely error made by the other student.

10. Identify, then correct, any errors in the student work below. How do you think the errors occurred?

! c - z f - r f + l - l f x ? j U \C-Z1

- -11 - K - l ) x ?

= 11 x ? —1

— n - '

Mid-Unit Review 6 9

Start Where You

Are

What Strategy Could I Try?

Suppose I have to evaluate this expression:

32(5° + 2 + 2 2 )

2(5 + 4 2 )

>• What math tools could I use? • mental math • mental math, and paper and pencil • a calculator

If I use only mental math, I might forget the numbers, so I write down the values of the numerator and denominator.

If I use mental math, and paper and pencil, • I must use the order of operations. • The fraction bar acts like a bracket, so I work on the numerator

and denominator separately. • I write down the values of the numerator and the denominator. • I look for friendly numbers to help with the division.

So, 32(5° + 2 + 2 2 ) 3 X 3 X 7

2(5 + 4 2 ) 2 X 3 X 7 3 2


If I use a calculator, • The fraction bar means divide the numerator by the denominator. • My calculator uses the order of operations. • Can I enter the expression as it is written? • Do I need to add extra brackets, or change any operations?

My calculator uses the order of operations, so I don't need any extra brackets in the numerator. The denominator is the product of 2 factors, so I do need to place brackets around these factors.

didn't use extra brackets. I realized that the numerator must be divided by both factors of the denominator. 1 divided by 2, then I divided by (5 + 42) 1

VI

Check Use any strategies you know to evaluate these expressions.

32 X 62 l . a )

d)

2. a)

i2 v *2 32 X 62 32 + 62

2 + 1

32 - 62

2 2 - 1

34 - 22

b)

e)

22

32

1

42 - 31 b)

42(34 -5- 2°) 2 (4 - 2 )

24(34 - 2°) 3(34 + 22)

Start W h e r e You Are

You wil l need • two different-coloured

number cubes labelled 1 to 6

Number of Players • 3 or more

Goal of the Game • To use the order of

operations to write an expression for the target number

How to Play t. Decide which number cube will represent the tens and which

will represent the ones of a 2-digit number. 2. One player rolls the cubes and states the 2-digit target

number formed. 3. Use three, four, or five operations.

Each player writes an expression equal to the target number. A power counts as an operation; brackets do not.

4. Score 1 point if you were able to write an expression that equals the target number. Score another point if you wrote a correct expression that no one else wrote.

5. The next player repeats Step 2, then all of you repeat Steps 3 and 4.

6. The first player to get 10 points wins.

72 UNIT 2: Powers and Exponent Laws

Exponent Laws I

FOCUS • Understand and apply

the exponent laws for products and quotients of powers.

When we multiply numbers, the order in which we multiply does not matter. For example, (2 X 2) X 2 = 2 X (2 X 2) So, we usually write the product without brackets: 2 X 2 X 2

Investigate

You will need 3 number cubes: 2 of one colour, the other a different colour

Two of you investigate multiplying powers. Make a table like this:

Product of Powers

54 X 52

Product as Repeated Multiplication

(5 X 5 X 5 X 5) X (5 X 5)

Two of you investigate dividing powers. Make a table like this:

Quotient of Powers Quotient as Repeated Multiplication

5 ' 5 ^ 5 < 5 5 / 5

Product as a Power

5?

Quotient as a Power

5'

Roll the cubes. Use the numbers to create powers, as shown.

Record each quotient of powers with the greater exponent in the dividend (the numerator).

Express each power as repeated multiplication, and then as a single power. Repeat the activity at least five times.

Use this number as the base

Use these numbers as the exponents

2.4 Exponent Laws I

Describe the patterns in your table. Share your patterns with the other pair in your group. How are your patterns the same? How are they different? Check your patterns with those of another group. Use your patterns to describe a way to multiply two powers with the same base, and a way to divide two powers with the same base.

WFF ""R " " < •• • ••• ' I

Connect

Patterns arise when we multiply and divide powers with the same base.

>- To mul t ip ly ( — 7)3 X (—7)5:

( - 7 ) 3 X ( —7)5 = ( —7)( —7)( —7) X ( - 7 ) ( - 7 ) ( - 7 ) ( - 7 ) ( - 7 )

= ( - 7 ) ( - 7 ) ( - 7 ) ( - 7 ) ( - 7 ) ( - 7 ) ( - 7 ) ( - 7 ) = ( " 7 ) 8

The base of the product is —7. The exponent is 8. The sum of the exponents of the powers that were multiplied is 3 + 5 = 8. This relationship is true for the product of any two powers with the same base. We use variables to represent the powers in the relationship:

I Exponent Law for a Product of Powers To multiply powers with the same base, add the exponents. am Xan = am+n

The variable a is any integer, except 0. The variables m and n are any whole numbers.

>- To divide 87 ^ 84: q7

87 - 84 = f r

_ 8 X 8 X 8 X 8 X 8 X 8 X 8 - 8 X 8 X 8 X 8

_ tfxrfxrfxrf X 8 X 8 X 8 ~ F x M

_ 8 X 8 X 8 ~ 1

= 8 X 8 X 8

= 8 3

So, 87 -h 84 = 83

The base of the quotient is 8. The exponent is 3. The difference of the exponents of the powers that were divided is 7 — 4 = 3. This relationship is true for the quotient of any two powers with the same base.

Divide the numerator and denominator of the fraction by their common factors: 8 x 8 x 8 x 8


I Exponent Law for a Quotient of Powers To divide powers with the same base, subtract the exponents. a m -5- a" = am ~ " m > n

a is any integer, except 0; m and n are any whole numbers.

rnmmm Simniif.ing Products and Quotients with the S a m . Base

Write each expression as a power. a) 6 5 X 6 4 b) ( —9)10 - ( ~ 9 ) 6

• A Solution

a) The powers have the same base. Use the exponent law for products: add the exponents. 6 5 X 6 4 = 6<5 + 4 )

= 6 9

b) The powers have the same base. Use the exponent law for quotients: subtract the exponents. (—9)10 -H (—9)6 = (—9)(1° "" 6)

= ( - 9 ) 4

? 7 ? H Evaluating Expressions Using Exponent Laws Evaluate. a) ( —2)4 X ( —2)7 b) 3 2 X 3 4 3 3

• Solutions

Method 1 Method 2

Simplify first using the exponent laws. Use the order of operations. a) The bases are the same. Add exponents. a) Evaluate each power first.

( —2)4 X (—2)7 = ( - 2 ) ( 4 + 7> Then use a calculator. = ( - 2 ) 1 1 ( —2)4 X ( —2)7 = 16 X ( - 1 2 8 )

= - 2 0 4 8 = - 2 0 4 8

b) All the bases are the same so add the b) Evaluate each power first. exponents of the two powers that are Then use a calculator. multiplied. Then, subtract the exponent of 3 2 X 3 4 H- 3 3 = 9 X 81 ^ 27

the power that is divided. Multiply and divide in order from left 3 2 X 3 4 -J- 3 3 = 3 ( 2 + 4 ) + 3 3 to right.

= 36 33 3 2 X 3 4 H- 3 3 = 27 = 3(6 - 3)

= 3 3

= 27

2.4 Exponent Laws I 7 5

Using Exponent Laws and the Order of Operations

Evaluate.

a) 62 + 6 3 X 6 2 b) ( - 1 0 ) 4 [ ( - 1 0 ) 6 -h ( - 1 0 ) 4 ] - 107

A Solution a) Multiply first. Add the exponents.

62 + 6 3 X 62 = 62 + 6 ( 3 + 2)

= 62 + 65 Evaluate each power. = 36 + 7776 Then add. = 7812

b) Evaluate the expression in the square brackets first. Divide by subtracting the exponents. (—10)4[( —10)6 4- (—10)4] - 107 = ( - 1 0 ) 4 [ ( - 1 0 ) ( 6 _ 4 ) ] - 107

= ( —10)4( —10)2 — 107 Multiply: add the exponents = ( -10 ) ( 4 + 2) - 107

= ( —10)6 — 107 Evaluate each power. = 1 000 000 — 10 000 000 Then subtract. = - 9 000 000

1. Use your own words to explain how to: a) multiply two powers with the same base b) divide two powers with the same base

2. Do you think it makes sense to simplify an expression as much as possible before using a calculator? Explain.

3. When can you not add or subtract exponents to multiply or divide powers?

D i s c u s s

t h e i r

r

Practice

Check 4. Write each product as a single power,

a) 5 5 X 54 b) 102 X 1011

c) (—3)3 X (—3)3 d) 216 X 214

e) ( - 4 ) 1 X (—4)3 f) 6 1 2 X 6 3

g) 2° X 24 h) ( —7)3 X ( - 7 ) °

5. Write each quotient as a power. a) 4 5 H- 4 3

c) 1510 h- 15°

x 2ÏL e) 2io

g) 65

b) 8 9 8 6

d) ( - 6 ) 8 - (—6)3

( -10) 1 2 f)

h)

( - 1 0 ) 6

( ~ i ) 5

( - i ) 4


Apply 6. a) Evaluate,

i) 3 4 -h 3 4

58

iii)

ii) ( - 4 ) 6

(-6)3

( - 4 ) 6

iv) 58 (-6)3

b) Use the results of part a. Explain how the exponent law for the quotient of powers can be used to verify that a power with exponent 0 is 1.

7. a) Compare these products. i) 3 4 X 3 9 ii) 3 9 X 3 4

b) Explain the results in part a.

8. Express as a single power. a) 3 4 X 3 9 -h 3 1 1

b) (—4) 3 H- ( —4)2 X (—4) 1 0

c) 6° X 6 3 -s- 6 2

43 X 45 (-3)4 x (~3)4

42 X 46 (-3)4 d)

9. a) Express as a single power, then evaluate. i) ( - 6 ) » X ( - 6 ) 7 - ( - 6 ) 7

ii) (—6)7 - ( - 6 ) 7 X (—6) 1

b) Explain why changing the order of the terms in the expressions in part a does not affect the answer.

10. Simplify, then evaluate. a) 102 X 102 + 104

c) 1011 - 103 X 106

e) 106 -h 102 X 102

1012

9)

i)

106

10" 104 X 102

b) 103 X 103 - 103

d) 101 + 105 X 102

f) 109 -5- 109

104 X 103 h)

102

j) 1$ + io2

11. a) Evaluate: 26 — 22 X 23

Describe the steps you used. b) Evaluate: 26 X 22 — 23

Describe the steps you used. c) Were the steps for parts a

and b different? Explain.

12. Assessment Focus An alfalfa field is a rectangle 104 m long and 103 m wide.

a) Write an expression for the area of the field, then evaluate the expression.

b) Write an expression for the perimeter of the field, then evaluate the expression.

c) i) Use the area in part a. Find all possible dimensions for a rectangular field with side lengths that are powers of 10.

ii) Find the perimeter of each field in part i.

d) Explain why the exponent laws are helpful for solving area problems, but not for perimeter problems.

13. Evaluate. a) 2 3 X 22 - 2 5 X 2

b) 3 2 X 3 + 22 X 2 4

c) 42 - 3° X 3 + 23

d) ( —3)6 h- ( —3)5 — ( —3)5 -i- ( —3)3

e) ( - 2 ) 4 [ ( - 2 ) 5 - ( - 2 ) 3 ] + (—2)4

f) - 2 4 ( 2 6 4- 22) - 2 4

9) ( - 5 ) 3 - ( - 5 ) 2 X ( - 5 ) ° + ( - 5 ) 2 - ( - 5 )

14. Provide examples to show why the exponent laws for products and quotients cannot be applied when the powers have different bases.

2.4 Exponent Laws I 7 7

15. Identify, then correct any errors in the student work below. Explain how you think the errors occurred.

16. Muguet uses a microscope to view bacteria. The bacteria are first magnified 102 times. This image is then magnified 101 times. a) Use powers to write an expression for the

total magnification. b) How many times as large as the actual

bacteria does the image appear?

17. a) Evaluate. i) 52 + 53 ii) 52 X 53

b) In part a, explain why you could use an exponent law to simplify one expression, but not the other.

18. a) Evaluate. i) 43 - 42 ii) 43 H- 42

b) In part a, explain why you could use an exponent law to simplify one expression, but not the other.

19. Simplify, then evaluate only the expressions with a positive value. Explain how you know the sign of each answer without evaluating.

a) (—2)2 X (—2)3

b) ( - 2 ) ° X (—2)5

c) (—2)5 - ( - 2 ) 3

d) (—2) 6 -f- (—2) 6

(—2)3 X (—2)4

( - 2 ) 3 X ( - 2 ) 2

( ~ 2 ) 6

e)

f) ( - 2 ) 3 X (—2)2

Take It Further 20. Find two powers that have a product

of 64. How many different pairs of powers can you find?

21. Write a product or quotient, then use the exponent laws to find the number of: a) centimetres in 1 km b) millimetres in 1 km c) kilometres in 105 m d) metres in 109 mm

22. Write a product or quotient, then use the exponent laws to find the number of: a) square metres in 102 km2

b) square metres in 106 cm2

c) square millimetres in 106 cm2

d) square centimetres in 1 km2

23. Explain how the exponent laws help you to convert among units of measure.

When can you use the exponent laws to evaluate an expression with powers? When can you not use these laws? Include examples in your explanation.

\


]> 1

FOCUS • Understand and

apply exponent laws for powers of: products; quotients; and powers.

A power indicates repeated multiplication. What is the standard form of (23)2? How did you find out? (2 3) 2 is a p o w e r of a p o w e r .

The base of a power may be a product; for example, (2 X 3)4. (2 X 3) 4 is a p o w e r of a p r o d u c t .

Investigate

Copy and complete this table. Choose your own power of a power to complete the 5th and 6th rows. Choose your own power of a product to complete the 11th and 12th rows.

O

Power!

lllfll ( 24)3

(32)4

[(-4)3]2

[(-5)3]5

(2 x 5)3

(3 x 4 f (4 x 2)5

(5 x 3)4

As Repeated Multiplication

24 x 24 x 24

(2 x 5) x (2 x 5) x (2 x 5)

As a Product of Factors

(2) (2) (2) (2) x (2) (2) (2) (2) x (2) (2) (2) (2)

2 x 2 x 2 x 5 x 5 x 5

As a Power Asa Product of Powers

2? x 5?

2.5 Exponent Laws II 7 9

What patterns do you see in the rows of the table? Compare your patterns with those of another pair of classmates. Use these patterns to record a rule for:

• writing the power of a power as a single power • writing the power of a product as a product of two powers

How can you check your rules?

We can use the exponent laws from Lesson 2.4 to simplify powers written in other forms.

>- Power of a power We can raise a power to a power. For example, 32 raised to the power 4 is written as (32)4. (32)4 is a power of a power.

(32)4 means 32 X 32 X 32 X 32. So, 32 X 32 X 32 X 32 = 32 + 2 + 2 + 2 Using the exponent law for the product of powers

>- Power of a product The base of a power may be a product; for example, (3 X 4)5. (3 X 4)5 is a power of a product.

(3 X 4)5 means (3 X 4) X (3 X 4) X (3 X 4) X (3 X 4) X (3 X 4) So, (3 X 4) X (3 X 4) X (3 X 4) X (3 X 4) X (3 X 4)

= 3 X 4 X 3 X 4 X 3 X 4 X 3 X 4 X 3 X 4 Removing the brackets = (3 X 3 X 3 X 3 X 3) X ( 4 X 4 X 4 X 4 X 4 ) Grouping equal factors = 3 s X 45 Writing repeated multiplications as powers

= 38

The exponent of 38 is the product of the exponents in (32)4. That is, (32)4 = 32 x 4

= 3 8

We can use this result to write an exponent law for the power of a power.

I Exponent Lavtf for a Power of a Power To raise a power to a power, multiply the exponents. (,am)n = amn

a is any integer, except 0. m and n are any whole numbers.

mn means m X n


We can use this result to write an exponent law for the power of a product.

I Exponent Law for a Power of a Product (ab)m = ambm

a and b are any integers, except 0. m is any whole number.

>• Power of a quotient 3

The base of a power may be a quotient; for example, I

5\3

g is a p o w e r of a q u o t i e n t .

| ) ' m e a „ s ( | ) x ( | ) x ( |

So- ( I ) x ( I ) * ( I ) = I * I x !

= 6 * 6 X 6 Multiplying the fractions

= ^ Writing repeated multiplications as powers

We can use this result to write an exponent law for the power of a quotient.

I Exponent Law for a Power of a Quotient

( ! ) " - £ a and b are any integers, except 0. n is any whole number.

We can use these exponent laws to simplify or evaluate an expression.

Simplifying a Power of a Power

Write as a power.

a) [ ( ~ 7 ) 3 ] 2 b) — (2 4) 5 c) (62)7

A Solution Use the exponent law for a power of a power. a ) [ ( ~ 7 ) 3 ] 2 = ( - 7 ) 3 x 2 b ) - ( 2 4 ) 5 = - ( 2 4 X 5) c) (62)7 = 6 2 x 7

= { - I f = — 22 0 = 6 1 4


Evaluating Powers of Products and Quotients Evaluate. a) [ ( - 7 ) X 5] 2 b) [24 - ( - 6 ) ] 4 c) - ( 3 X 2)2 d) ( G ) '

• Solutions

Method 1 Method 2

a) Use the exponent law for a power of a product. [ ( - 7 ) X 5] 2 = ( — 7)2 X 5 2

= 49 X 25

= 1225

Use the order of operations. a) [ ( - 7 ) X 5]2 = ( - 3 5 ) 2

= 1225

b) Use the exponent law for a power of a quotient. Write the quotient in fraction form.

b) [24 - (—6)] 4 = (—4) 4

= 256

[24 (—6)] 4 = ( ^ ) 4

_ 244

( ~ 6 ) 4

_ 331 776 1296

= 256

c) Use the exponent law for a power of a product. - ( 3 X 2) 2 = — (32 X 2 2 )

= - ( 9 X 4)

= - 3 6

c) - ( 3 X 2) 2 = - ( ô ) 2

= — 6 2

= - 3 6

d) Use the exponent law for a power of a quotient. / 78 \ 3 _ 783

\ 1 3 / 133

d. ( S ) ' - 63

= 216

_ 474 552 2197

= 216

We can use the order of operations with the exponent laws when an expression involves the sum or difference of powers.


Applying Exponent Laws and Order of Operations

Simplify, then evaluate each expression. a) (32 X 33)3 - (43 X 42)2 b) (6 X 7)2 + (38 h- 36)3 c) [(~5)3 + ( -5 ) 4 ] 0

A Solution

Use the exponent laws to simplify first, where appropriate. a) In each set of brackets, the bases are the same, so use the exponent law for products.

(32 X 33)3 - (43 X 42)2

= (32 + 3)3 — (43 + 2)2 Add the exponents in each set of brackets. = (35)3 — (45)2 Use the power of a power law. = 35 x 3 — 45 x 2 Multiply the exponents. = 315 — 410 Use a calculator. = 14 348 907 - 1 048 576 = 13 300 331

b) Multiply in the first set of brackets. Use the exponent law for the quotient of powers in the second set of brackets. (6 X i f + (38 -5- 36)3

= (42)2 + (38 _ 6)3

= 422 + (32)3 Use the power of a power law. = 422 + 36 Use a calculator. = 1764 + 729 = 2493

c) The expression is a power with exponent 0, so its value is 1. [ ( -5 ) 3 + ( -5 ) 4 ] 0 = 1

1. Why do you add the exponents to simplify 32 X 34, ^ but multiply the exponents to simplify the expression (32)4?

2. a) What is the difference between a quotient of powers and a power of a quotient?

b) What is the difference between a product of powers and a power of a product?

3. In Example 3, is it easier to key the original expressions in a calculator or use the exponent laws to simplify first? Justify your answer.


Practice

Check 4. Write each expression as a product of

powers. a) (6 X 4) 3 b) (2 X 5) 4 c) [ ( - 2 ) X 3] 5

d) (25 X 4 ) 2 e) (11 X 3)1 f) [ ( - 3 ) X ( - 2 ) ] 3

5. Write each expression as a quotient of powers.

a) ( 8 - 5 ) 3 b ) (21 - 5)4 c) [ ( - 1 2 ) - ( - 7 ) ] 5

« ( f ) ' « ( } ) ' «

6. Write as a power. a) (3 2 ) 4 b) (6 3 ) 3 c) (5 3 ) 1

d) (7 0 ) 6 e) — (8 2) 2 f) [( —3)4]2

7. Simplify (24)2 and (22)4. What do you notice? Explain the results.

8. Write each expression as a product or quotient of powers. a) [3 X ( —5)]3 b) - ( 2 X 4 ) 5

« ( ! ) ' « ( 3 ) ' e) - [ ( - 1 0 ) X 3] 3 f) (16 -H 9) 2

Apply 9. Why is the value of ( — 52)3 negative?

10. Simplify each expression, then evaluate it. For each expression, state the strategy you used and why.

a) (3 X 2) 3 b) [ ( - 2 ) X 4] 2 c) ( ^ ) 3

d) ( f ) 2 e) (12 8 ) 0 f) [ (—4) 2 ] 2

11. Why is the value of [(—2)3]4 positive but the value of [(—2)3]5 is negative?

12. Compare the values of — (42)3, (—42)3, and [(—4)2]3. What do you notice? Explain the results.

13. Assessment Focus For each expression below: i) Evaluate it in two different ways:

• do the operation in brackets first • use the exponent laws

ii) Compare the results. Which method do you prefer? Was it always the same method each time? Explain.

a) (4 X 3) 3 b) [ ( - 2 ) X ( —5)]2 c) ( § ) '

d) ( y ) ° e) [(~5)2]2 f) (25)3

14. Simplify, then evaluate. Show your work, a) (32 X 3 1 ) 2 b) (46 -h 4 4 ) 2

c) [ ( - 2 ) ° X (—2) 3 ] 2 d) (10 6 -f- 104)3

e) (10 3 ) 2 X (10 2 ) 3 f) (12 2 ) 4 - (12 3 ) 2

g) (5 2) 6 - (5 3 ) 4 h) [ ( - 2 ) 2 ] 3 X ( - 2 ) 3

15. Find any errors in this student's work. Copy the solution and correct the errors.

8 4 UNIT 2: Powers a n d Exponent Laws

16. Simplify, then evaluate each expression. a) (42 X 4 3 ) 2 - (54 4- 5 2 ) 2

b) (33 3 2 ) 3 + (84 X 8 3 ) 0

c) (23)4 + (24 + 2 3 ) 2

d) (62 X 6 0 ) 3 + (26 - 2 4 ) 3

e) (53 X 53)0 - (42)2

f) (105 H- 102)2 + (33 H- 3 1 ) 4

17. Simplify, then evaluate each expression. a) [(—2)3 X (—2)2]2 - [(—3)3 - (—3)2]2

b) [(—2)3 -T- (—2)2]2 - [(—3)3 X (—3)2]2

c) [(—2)3 X (—2)2]2 + [(—3)3 - ( - 3 ) 2 ] 2

d) [(—2)3 (—2)2]2 + [( —3)3 X (—3)2]2

e ) [ ( — 2 ) 3 -h ( — 2 ) 2 ] 2 — [ ( — 3 ) 3 -r- ( — 3 ) 2 ] 2

f) [(—2)3 X (—2)2]2 + [(—3)3 X (—3)2]2

18. Use grid paper. For each expression below: i) Draw a rectangle to represent the

expression. ii) Use the exponent laws to write the

expression as a product of squares. iii) Draw a rectangle to represent the

new form of the expression. iv) Compare the two rectangles for each

expression. How are the rectangles the same? How are they different? Use these rectangles to explain how the square of a product and the product of squares are related,

a) (2 X 3)2 b) (2 X 4) 2

c) (3 X 4) 2 d) (1 X 4) 2

19. Simplify, then evaluate each expression. a) (23 X 2 6 ) 2 - (37 -s- 3 5 ) 4

b) (6 X 8)5 + (53)2

c) [(—4)3 X ( —4)2]2 + (43 X 4 2 ) 2

d) [( —2)4]3 + [ ( — 4 ) 3 ] 2 - [( —3)2]4

e) [ ( — 3 ) 4 ] 2 X [(—4)0]2 — [(—3)3]0

f) [ ( - 5 ) X ( - 4 ) ] 3 + [ ( - 6 ) 3 ] 2

- [(—3)9 -H (—3)8]5

Take It Further 20. a) Write 81:

i) as a power of 9 ii) as a power of a product iii) as a power of 3

b) Write 64: i) as a power of 8 ii) as a power of a product iii) as a power of 2

c) Find other numbers for which you can follow steps similar to those in parts a and b.

21. a) List the powers of 2 from 2° to 212 in standard form.

b) Use your list from part a to write each number in the expressions below as a power of 2. Evaluate each expression using the exponent laws and the list in part a.

i) 32 X 64

iii) 1024 h- 128

v) (8 X 4) 3

ii) 16 X 8 X 32 16 X 256 iv) 1024

4

* (fr )

Design and create a poster that summarizes all the exponent laws you have learned. Provide an example of each law.


Study Guide

I A power represejits repeated multiplication. 25 = 2 X 2 X 2 X 2 X 2

= 32

( - 3 ) 4 = ( - 3 ) ( - 3 ) ( - 3 ) ( - 3 )

= 81

- 3 4 = - (3 ) (3 ) ( ]3 ) (3 )

= - 8 1

I A power with an integer base, other than 0, and an exponent 0 is equal to 1. 2° = 1

( - 4 ) ° = 1 - 4 ° = - 1

I To evaluate an expression, follow this order of operations: Evaluate inside brackets. Evaluate powers. Multiply and divide, in order, from left to right. Add and subtract, in order, from left to right.

Exponent Laws m and n are whole numbers. a and b are any integers, except 0.

I Product of Powers amX a" - am + n

• Quotient of Powers am -r a" — am ~ n m > n

I Power of a Power (,am)n = amn

> Power of a Product (ab)m = ambm

I Power of a Quotient

(f)" = ¥ 8 6 UNIT 2: Powers and Exponent Laws

Review

1. Write as repeated multiplication, then in standard form. a) 4 3 b) 72 c) —( —2)5

d) — 3 4 e) - l 8 f) ( - 1 ) 8

2. Use tiles and cubes to explain the difference between 22 and 23.

3. Write as a power, then in standard form. a) 3 X 3 X 3 X 3 X 3 X 3

b) (—8)( —8)(—8)

c) - ( 2 X 2 X 2 X 2 X 2 X 2 X 2 )

d) 12 X 12

e) 4 X 4 X 4 X 4 X 4

f) (—5)(—5)( —5)(—5)

4. Explain the difference between 58 and 85.

5. A telephone tree is used to send messages. The person at the top calls 2 people. Each person calls 2 more people. Suppose it takes 1 min to call someone. A message is relayed until the bottom row of the tree has 256 people. How long does this take? How do you know?

6. a) Is the value of — 42 different from the value of (—•4)2? What purpose do the brackets serve?

b) Is the value of — 23 different from the value of (—2)3? What purpose do the brackets serve?

7. a) Evaluate each expression. i) — 32 ii) — (3)2 iii) —(—3)2 iv) ( - 3 ) 2

b) For each expression in part a that includes brackets, explain what the brackets show.

8. Write as a power of 10. a) one hundred million b) 10 X 10 X 10 X 10 c) 1

d) 1 000 000 000

e) one thousand

9. Use powers of 10 to write each number. a) 700 000 000 b) 345 c) 80 027

10. a) Copy and complete this table.

Power Repeated Standard Multiplication Form

35 3 x 3 x 3 x 3 x 3 243

34

3 x 3 x 3

32

3

b) Describe the patterns in the table. c) Extend the pattern to show why any

number with an exponent of 0 is equal to 1.

Review 8 7

11. a) The tallest tree in the world, Hyperion in California, is about 102 m tall. The highest mountain, Mount Everest, is about 104 m high. About how many times as high as the tree is the mountain?

b) Earth's diameter is about 107 m. The largest known star has a diameter of about 1012 m. About how many times as great as the diameter of Earth is the diameter of the largest known star?

12. Write each number in standard form. a) (4 X 103) + (7 X 102) + (2 X 101)

+ (9 X 10°)

b) (3 X 105) + (2 X 102) + (8 X 10°)

13. Evaluate, a) 3 4 + 32

c) 103 - 102

b) (—4)2 + ( - 4 ) 3

d ) ( - 5 ) 4 - ( ~ 5 ) 2

14. Evaluate. a) 2 3 + (5 - 2 ) 4

b) 100 -H 2 + (4 + l ) 3

c) (62 + 7 2 ) 0 - (84 + 2 4 ) 0

d) 3 X 23 + 8 h- 4

e) (21 H- 7) 4 - 2 3

f) [ ( - 4 ) ° X 10]6 H- (15 - 10)2

15. Scientists grow bacteria. This table shows how the number of bacteria doubles every hour.

Time ! Elapsed Time Nijmberof I After Noon (h) Efacteria

noon 0 1000 x 2° 1 : 0 0 P.M. 1 1 0 0 0 X 2 1

2 : 0 0 P.M. 2 1 0 0 0 X 2 2

3 : 0 0 P.M. 3 1 0 0 0 X 2 3

a) Evaluate the expressions in the table to find the number of bacteria at each time.

i) noon ii) 1:00 P.M. iii) 2:00 P.M. iv) 3:00 P.M.

b) The pattern continues. Write an expression, then evaluate it, to find the number of bacteria at each time.

i) 4:00 P.M. ii) 6:00 P.M.

iii) 9:00 P.M. iv) midnight

16. Use a calculator to evaluate this expression: 4 3 - (2 X 3) 4 + 11

Change the position of the brackets. Evaluate the new expression. How many different answers can you get by changing only the position of the brackets?

17. Identify, then correct, any errors in the student work below. Explain how you think the errors occurred.

s (-2)Z x ^ ^ ^

i - ( -z f -1 t-li-..

! - -*?2 - •+- 16

I \C

t


18. Write each product as a power, then evaluate the power. a) 5 3 X 54 b) (—2)3 X ( - 2 ) 2

c) 3 2 X 3 3 X 31 d) - 1 0 4 X 10°

19. There are about 1011 galaxies in the universe. Each galaxy contains about 10n stars. About how many stars are in the universe?

20. Write each quotient as a power, then evaluate the power. a) 75 -T- 73 b) ( - 1 0 ) 9 h- ( - 1 0 ) 3

67 C) £ d)

21. a) Can you use the laws of exponents to simplify 63 X 55? Explain.

b) Can you use the laws of exponents to simplify 272 92? Explain.

22. Find and correct any errors in the student work below. Explain how you think the errors occurred.

= i-v?

1 - -27

! c ) tWISL - t®--

V>) (~*f +* ^ ^ tr-VA

23. Write each expression as a product or quotient of powers, then evaluate it. a) (3 X 5)3 b) (12 - 3)5

c) [ ( - 4 ) X 2]4 d) (63 X 44)°

« ( I « 15

24. Write each expression as a power. a) (32)3 b) (40)6

c) [(—2)3]3 d) (55)2

25. For each expression below: Evaluate it in two different ways:

i) do the operation in brackets first ii) use the exponent laws

In each case, which method is more efficient? Explain why. a) (5 X 3)3

b) (3 X 3)4

c) (8 - 2)5

• (I) ' e) (23)4

f) (62)0

26. Write each expression as a power, then evaluate. a) 6 4 X 6 3

b) ( — l l ) 7 — C — l l ) 5

, 34 X 3 5

3

5 5

d) , 7 5 X 5

e) ( — 4 ) 3 X ( — 4) 6

( — 4) 2 X ( — 4 ) 4

106 X 10° 103 X 102

27. Simplify, then evaluate each expression. a) 23 X 22 - 2° + 2 4 -r 23

b) ( - 2 ) 3 X (-2Y

( - 2 ) 3 - ( —2) 2

c) 122 X 124 -5- (—2)4 - 12°

( - 1 2 ) 2 X ( - 1 2 ) 4

d) ( - 2 y 12

Review 8 9

Practice Test

1. Write as a product or quotient of powers.

a) (3 X 4) 3 b) [ ( - 5 ) X 2] 4 « U d) - ( f

2. Simplify. a) ~ ( 2 3 ) 3 b) (6 2 ) 0

3. Simplify each expression, then evaluate it.

a) [ ( - 3 ) X (—2)] 4 b ) ( i

0 [ ( - 5 )

0 (6 0) 4

213 d) - [ ( - 3 ) 2 ] 4

d) [ ( - 3 ) 2 ] 3

4. Is the value of a power with a negative base always negative? Or, is it always positive? Or, is it sometimes negative and sometimes positive? Illustrate your answer with some examples.

5. A baseball diamond is a square with side length about 27 m. Is the area of the baseball diamond greater or less than 103 m2? How do you know?

6. Explain why the brackets are not necessary in this expression: ( —35 X 10) - (9 3)

Evaluate the expression, showing each step.

7. Identify the correct answer for (23 + 4)2 X ( —10)3 (5 + 5)2. a) - 2 4 0 b) - 1 4 4 0 c) 1440 d) - 2 8 825

Explain how each of the other incorrect answers could have been determined.

8. Evaluate only the expressions with a positive value. Explain how you know the sign of each expression before you evaluate it. a) ( ~ 5 ) 3 X ( —5)2 - ( —5)1 b) [( —9)6 ~ ( - 9 ) 3 ] 0

i ( ~ 1 ) 2 X ( - 1 ) 4

(— l ) 3 X (— l ) 2 d) (—4) 6 + ( —4)4 X ( - 4 ) °


Unit P rob lem How Thick Is a Pile of Paper?

You will need a sheet of paper and a ruler.

>- Fold the paper in half to form 2 layers. Fold it in half again. Keep folding until you cannot make the next fold.

>• Create a table to show how many layers of paper there are after each fold.

Complete the table for the number of folds you were able to make.

>- Look for a pattern in the numbers of layers. How can you express the pattern using powers? Draw another column on your table to show the Number of Layers as Powers. Suppose you could make 25 folds. Use patterns in the table to predict a power for the number of layers after 25 folds. Evaluate the power.

>- Measure the thickness of 100 sheets (200 pages) in your math textbook. Use this measure to calculate the thickness of 1 sheet of paper in millimetres. How high would the layers be if you could make 25 folds? Give your answer in as many different units as you can. What do you know that is approximately this height or length?

Your work should show: • a completed table showing the numbers of layers • the calculations of the thickness of 1 layer and the height after 25 folds • an example of something with the same height

Reflect o n Y o u r Learning What have you learned about powers and

their exponent laws? What ways can you think of to remember the laws and how to use them?

Unit Problem

powers and exponent laws - ms. johnsonsolve problèms involving powers. • perform operations with...

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